CONVOLUTION-TYPE
INTEGRAL EQUATIONS
IN T H E C L A S S OF G E N E R A L I Z E D F U N C T I O N S V. B.
Dybin
UDC517.948.32/33
a n d N. K. K a r a p e t y a n t s
An extensive class of boundary-value problems for partial differential equations can be formulated in terms of the Wiener- Hopf equation
Z/(z) +
k(z -- t)l(t)dt = g(x),
z > O,
(1)
or in terms of its generalizations, in particular, the "dual" equations
~,iJ(x)~--~-n~_kt(x--t)J(t)dt=g(x), x> O, (2) ~,zl(x) q-
kz(x -- t)l(t)dt = g(x),
x < O,
(see [1,2] for example). Until recently solutions if(x) of the above equations, depending on the properties of the coefficients kj(x) and g(x), were mainly sought in various classes of either smooth or summable functions. The lack of a sufficiently rigorous proof of the applicability of the Wiener-Hopf method led to a series of articles ([3,4] and others)in which Eq. (2) was completely investigated for several fundamental cases by methods from the theory of boundary-value problems and singular integral equations, and in particular by the Nemann method. A complete qualitative investigation of the normal case of Eq. (1) by functional-analytic methods is given in [5]. A modification of the Wiener-Hopf method has been used in finding solutions of Eqs. (1) or (2) in the class of hounded functions in connection with applied problems. A factor e-elxl with e > 0 small is introduced on the right side of the equation, and the resulting modified equation is solved by means of successive limiting operations with respect to 8; this method requires further justification.* In various applied problems such as certain contact problems in elasticity the need has recent2y arisen to study Eq. (1) in the class of slowly increasing functions. On the other hand a series of problems in automatic-control theory and some other areas (see [7] and the articles cited therein) have been reduced to the Wiener-Hopf equation with solutions in the classof g.f. (generalized functions) of finite order. In the present work we propose a general scheme for the solution of the above equations with solutions in the class of slowly increasing g.f. of finite order. A special case of these classes is encountered in spaces of functions of the form (x + i)kif(x), where k -> 0 is an integer and :f(x) is any function that is either bounded or square summable. Our investigation uses the method of the Riemann problem with coefficients that can have integral-order zeros at finite points of the contour. The problem of solving Eq. (2) in the class of g.f. was apparentIy first posed in [8]. In the normal case the equations considered below were investigated in [9] in classes of g.f. close to those we consider, and in one special case they were considered in the same classes in [10] (see also [11]). *This case was completely investigated in a recently published article [6] (see also [5], Sec. 8). Translated from SibirsMi Matematicheskii Zhurnal, Vol. 7, No. 3, pp. 531-545, May-June, 1966. OAginal article submitted May 17, 1965.
429
1.
Classes
of Solutions.
Fourier
Transforms
As a basis for the method of solving Eqs. (1) and (2) by means of the Riemarm problem in classes of generalized functions we use the following facts (see [4] for example): firstly, the Fourier transform of a function vanishing identically for x < 0 (x > 0) is analytic in the upper (lower) half-plane; secondly, the Fourier transform of the convolution of two functions is the product of their Fourier transforms. Fourier transforms have similar properties in wide classes of g.f., and so the method referred to can be successfully applied in the solution of integral equations in these classes. This was first clearly demonstrated by Yu. I. Cherskii [9]. In this section we define the classes we will investigate, we investigate some of their properties, and we derive the apparatus for reducing the integral equations to the Riemarm problem in the class of g.f. 1. Spaces of test functions and generalized fu.nctiorm.. For the space of test functions we use one of the spaces Lp{-~; --m} (p = 1,2,m -> 0 an integer) of functions ~(x) defined on the real axis, m times differentiable, and satisfying the conditions
d~ xk---~-x~,(x)~Lv(--oo, oo),
] = O,t,...,m;
k--O,t,
2....
The topology of the s?ace is defined by means of a countable sequence of comparable and compatible norms:
--oo
The space Lp{-~o; --m} is a complete countably-normed space (see [12]) and it can be represented in the form co
L, {--oo;--m} = fl Lp {--k;--ra}, k=0
where
Lp{O; --m} ~Lp{--i; --m} ~ . . . ~ L p { - - k ; --m} ~...~Lp{--oo; --m}, Lp{-k; --m} is the completion of Lp{--~; --m} in the kth norm (3). We write Lp {~o; m} (p = 1,2) for the space of g.f. ] , i.e., the linear continuom functional defined on Lp{--~o; -m}. For Lp{~O; m} we have the representation
=
L, {k;m).
where Lp{k; m} is the conjugate of the normed space Lp{-k; --m}. 2. Fourier transforms. We write La{-m; --~} for the space of test functions O(x) which are Fourier transforms of elements ~0(t)EL2{--,o;-m}: t oo
r
Vq
-- _
The space Lz{-m; --~} is the complete countably-normed space of infinitely differentiable functions ~(x) for which
da
xJ.-d~x~ q)(x) ~ L 2 ( - - o o , oo),
] = O, t, . . . . m,
k ---~ O, t, 2, . . . ;
the topology of this space is determined by the countable sequence of norms
d~,
IMM,
k =
O, t, 2 . . . . .
--oO
We write Lz{m; .o} for the space of Fourier transforms of g . i of b2{,o; m}; the result of applying the Fourier operator F = Vf to an element f EL2{'~ ; m} is defined by the relation
(F, r 430
= (l, ,~(--x)), ~ ( ~ { - - o o ;
--m},
e=
V~,(~Zo.{--m;
--oo}.
The space L2{m; -} is the space of g.f. on the test space l~{--rn; --~}. The structures of Lz{--rn; -~} and L2{m;~o} are described by the relations co
L~ {-- m; --,oo} =
L= {m; oo} -~- U L: {m; k},
fl L~ 1-- m; -- k},
k~---0
h~----0 where h { em; ek} = V[Ie{ •
+m}].
In the sequel we will need THEOREM 1. Let the support N of the g.f. F~I~{0; -} consist of a finite number of isolated points, N = { a 1, ...,ap}. Then nh
F ~--- ~
~ , chj 6(J)(x - - ah),
where the Ckj are constants and n k is a fixed number. This theorem can be proved by the method used in [12] (pp. 149-150). We write R{-m; ---~} for the space of Fourier transforms of functions ~0(t)Eh{--~; -rn}. The space a{-m;-~} is a countably-normed complete space of infinitely differentiable functions ~(x) such that xmr -* 0 when Ix l-~ ~(J = 0,1,2 .... ). The topology of R{--m; --~} is generated by a sequence of norms introduced externally: ~l~lt k = I~~0{Ik (k = 0,1,2 .... ), where ~ = V~0 and ll~ollk has the form (3) for p = 1. We write R{m; -},for the space of g.f. defined on the test space R{--m; -co}or,equivalently,the space of Fourier transforms F = V~ of g.f.] ~Ll{m; ~}. The structure of R{ ~m; +-} can be described just as the structure of the spaces LI{+=, fin} was described above. Remark. We have formally defined two pairs of spaces of g.f. in which we will carry out our investigatiom. For each of these spaces we can prove theorems concerning the general form of a functional. For example any g.f~ FEIe{m; m} has the form F = (x + i)mdk(F0(x))/dxk, where F0(x) i~ a continuous function, and the operation of differentiation is to be taken in the sense of the differentiation of functionats. Similar facts hold in the other g.f: spaces introduced. 3. The spaces I~• ~o}, R:~{m; ,}. Using a definition of the space R{0; -~,} (Lz{0; --~}) equivalent to that used above: ~ (x)~R{0; --~} (Ie{0; --~}) if ~(k)(x) R{0; 0} = V[L~(--~, ~)] (L~{ 0; 0} = I~(--~, ~)), k = 0,1,2 ..... we show that the following 1emma holds. LEMMA. The Cauchy operator Sr =
1 ~ q)(t)dt z~i ~ t - - x
establishes a linear homeomorphism of the space
--oo
R{0; -~} (~{0; -*~) into itself. Proof. Let O(x) eR{0; --~}. If the function q(t) = V-1O belongs to the space Ll{--~; 0}, then so does the function q(t) sgnt. In particular ~o(t) sgnt6Li(---, ~o). According to a lemma in [13] the Fourier transform of this function is
i J ~ ~ (t) dt ~ R {0; 0}. Sq)---- V[(p(t)signt] ~--~-. ~i
t--x
O~
Since iiSOH~ =
I [(x+
~
00
i)~[sgnxq~(x)] [dx -~- [[OHk,
the first part of the 1emma is obvious. The
reasoning is similar for the space I~( 0; --~}. The 1emma shows that for any function ~(x)6Lz{0; --~}, we have the Sokhotskii formulas
r where ~ i(x) = + 89162 + 89 L2 ~[ 0; - ~ t .
= (I)+(z) - o - ( z ) ,
so(z)
= r
+r
We denote the spaces of test functions from Lz{O; --~} of the form r •
by
431
Then the following chain of relations defines the spaces La• Ix -t- i)-roPe- @s • {0; co}; F,:• @/e• oo}, if (Ft • O • It follows from the above definitions that for F Q Lz{0; co/
SF ~--- F+nt-F -, relation (SF, r
oo} : F • @Lee{m; co}, if F, e = - - 0, where * • @La:~{0; --oo}. the Sotdaotskii formulas F ~--- 17+ _ F - ,
also hold, where F • = -+-~F-~ -~SF, and the operator SF acting on g.f. is defined by the
= (F, - - S r
Similar constructions are also valid for the space R{m; ~ } . 4. The spaces Lp•
m} (p = 1,2). We write Lp•
--rn} for the spaces of test functions
{p• L n { - - c o ; --rn}, vanishing identically for x < 0 and x > 0, respectively. Then Lp+{co; m) are subspaces of g.f. ]_+ @Lp{co; m}, satisfying the conditions (f• q~) = 0, where qD~=(x).@Lp.~{--c~; --rn}. ra--1
THEOREM 2. If /Q Lp{oo; m} and ]e Q Ll~•176176 m}.
we have the representation f = ] + - ] .
+ )co where ~e = ~ Cg6(~l(x), k=o
One of the several proofs of this theorem can be found in [9]. We note that the functions are zero for x = 0 together with their derivatives up to the order m -- 1 inclusive.
(p• (x) 6_Lp• {--oo; --m}
m---1
Hence, if the g.f. A can be written in the form
]o -~- E C~6(k)(x)' we have ]0 Q Lp•
rn}.
The con-
k----q)
verse also holds: if g-f./0 @Lp•
m},then it is a linear combination of g.f. 6(k)(x) (k = 0,1 ..... m - 1).
The relation between the spaces Lp• m} and L~• co}, R• co}, i.e., between the g.f. f~, equal to zero on one of the semiaxes x < 0 or x > 0,and the g.f. F • "analytioally continuable" into the corresponding half-plane, is established by the following theorem. THEOREM 3. A necessary and sufficient condition that the g.f.f• @L~•
m}
(p = 1,2) is that
R e {rn; oo}, p = t, F• = VI• @ L2• {rn; oo}, p = 2. Proof. Necessity. For simplicity let m = 1. If ]+ @L2+{oo; t},
then (1+, (p_) = 0
(p_(z) ~/.r - - t } . Further let ~" = Vf+, i.e., Ft "-- (x --[- i)-tF + ~ L~{0; co}. 0, where ~+(x) is an arbitrary function from Lz+{0; --~}. In fact,
(F,, (1)§
) = (F+, (x -if- i)-'q)+(x) ) =
for all
We show that (F1, ~+) =
(t+, V-'[ (x + i)-'(1)+(x)] ( - - t ) ) .
Since -t
~(t) = V -1 [(x + i ) - i O + ( x ) ] ( - - t ) = Ce-t I e-xe~+(x) dx, O
where (p+(x) = V-'(I) + Q L2+{--co; 0}, Hence, (Ft, O+) = 0 and F + ~/~.+{1; co}.
it follows that ~ ( t ) ~--- 0
for t -> 0, i.e., r
(~ L~-{--co; - - ] } .
Sufficiency. Let F + Q L2+{t; co}, Le.,(F+, (x --~ i)-t(D+(x)) = 0,where (I)§ E L-z+{0; co). For any (p_(x) Q L2{-- co; --1} we have (V-IF+, ~=(x)) = (F+, V[~_] (-x)) = (F§ (x + i)'l[(x + i) O+(x)]), where
~ + ( x ) - - v[~_] ( - z ) 6 L~+{--t; - - ~ } .
Hence (z + 0 r
0 L2+{0;'--oo} and so (1+,~-) =
(V-1F§162 = 0. The theorem is proved. 2.
Coefficient
Classes.
Factorization
In the following we assume that the kernels kj(x) of the convolutions in Eqs. (1) and (2) belong to the space LI{--~;0}; hence their Fourier tramforms Kj(x) = Vkj are etements of the space R{0;--'~,},which is a commutative countably-normed ring with respect to ordinary multiplication. 1. The Wiener--Levy and Wiener Theorems. Following [5] (pp. 13-16), we write ~{0; --,~ (R'*{0; - - 4 ) for the countably-normed ring of functions obtained by extending the ring R{0; --,~ (R~{0; --*~) by one dimension by adjoining a constant. The following theorems W, L, and W~ can be transferred naturally to this ring.
432
THEOREM W. Let Q(z) be a function holomorphic in a region D located in general on a multisheeted Riemann surface, and let .g(x) ~ 1~{0; --oo} be such that the curve z = K(x) can be assumed to lie inside D. Then -
co).
Proof. The function Q ( g ) @/~{0; - o o } , if it and ait its derivatives belong to the ring /~{0; 0} ~- ~. Since J~ (x) 6 ~, it follows from Theorem W for the ring e that (2 ( g ) ~ ~. Moreover /~(I)(x) ~ ~, and Qa)(_~) 6 ~, and so d/dx[Q(l?)] = Q(i)(_~)/~'a)6 ~ etc. we can similarly prove Then
THEOREM L. Let k(x) 6 L i { - - o o , 0 } and ~ q- K(x) =/=0(--oo ~ x <~. oo) Ind (~.-}- K(x) ) = ' 0 . in (~ -}- K(x)) 6 1/{0; --oo}, where the appropriate branch of the logarithm is used.
THEOREM Wl. Let Q(z) be holomorphic in a region D containing alI the values of the function R• /~---{0; --co} in the upper (lower) half-plane. Then q(J~• 6 / ~ : { 0 ; --oo}.
Q
2. Factorization. By the factorization of a function /~(x) 6 R{0; --oo} /~(x) =/= 0 (--oo ~ x ~ oo), we mean its representation in the form
( x+i ;
=
;r -
i
x+(z)/X-(z),
where u ~- Ind ~ ( x ) , X ~:, [X• - ' 6 R-~{0; --oo}, Ind X+ =
Ind X - = 0.
Remark. In boundary-value problem theory another name is used for the operation: "the representation of a function K(x)in the form of a ratio of limiting values of canonical functions" (see [14], pp. 115-117, for example). We use the term "factorization" for brevity. Let ~t = 0. Then from Theorem L we have l n / ~ ( x ) 6 ~{0; tions of the following problem concerning discontinuities: In X+(x)
-
-
In X-(x)
=
--oo},
and the functions in X~(x) are solu-
In R ( x ) .
(5)
The unique solution of (5) in the ring a~{0; ---oo} is obtained from the Sokhoukii formutas
"- - c o
t--X
'
and it is easily shown that the desired representation (4) is given by the functions
X~(x) = exp F• In the general case,when ~t ~ 0, we multiply both sides of (4) by (x + i/x --i) ~t and then use the same method to obtain
x
x-k~)
t
R(~)}+-f~
X--~
3.
"Dual"
Equations
in t h e S p a c e
~
lnt\t--i
]
t --X
L2{*r ; m}
In seeking solutions of "dual" equations in the class of g.f., we interpret the conditions (2) in the following sense:
(7)
433
For given functions kj(x) ~ L~{--oo; 0} (J = L2), a g.f. g ~ ~ { o o ; . m } , and the required g.f. ], the relations (7) must be satisfied for any test functions q~• ~ s177 --m}. For the class of kernels kj ,indicated above, the convolution operations on the left in (7) are continuous in
z ~ { - ~ ; -m}. We extend the Eqs. (7) to the whole axis:
(], ~,icp(x) A- ~
~ k, (t -- x) ~ (t) dt) = (g+, (p) -{- (h-, (p), (8)
(1, ~uzq)(x) + - ~
kz(t -- x)cp(t)dt) --= (g_, ~p) + (h+, ~), --OD
where h• Q s177 m} are new unknown g.f. Taking the Fourier transform of (87, we obtain the following eqnivalent system of equations in the space L2{m; ~}: (97
[)~l -F K~(x) ]F -= G+ + H-,
[)~2 + Kz(x) ]Y = G- +1t+.
Let k I ~ 0 and k z ~ 0. We assume that the functions Xj + Kj(x) have a finite number of zeros on the real axis and can be written in the form
Li -~ Kl(x) -----A ( x ) C ( x ) ~ i ( x ) / (x -4- i) ~+~, ~ + K~(x) = B(x)C(x)$:2(x) / (x -- i)~(x + i)~,
p A(x)-=
l~
q (x--ak)%,
B(x)=
l~
r (x--bj)~,,
C(x)----- I ~ t ( x - c 0 , , ,
where a k ~ bj and c l are real numbers; ak, Sj, and Yl are positive integers, Yah ~ a, k
Kj(x) 6 R{0; - o o } ,
and ~ ( z )
E~j ~ j
~,
Y'Yt ~ y, t
~ 0, --oo ~< x ~< ~o, j = ~.~.
To ma~e the left-hand sides of the equations in (9) identical, we multiply them by B(xTKz(x7(x --i7 -B and A(x)Kl(x) (x + i) -a, respectively. We obtain the following riemann problem in the space Lz { m; *0}:
(z + i) ~ (g+ + a - ) = K(x)j where K(x7 = Kz(xT/~,l(x) E~{0; --~r
~(//- + a+),
we permit factorization according to formulas (47 and (6~
Multiplying both sides of (10) by ((x + i)mx+(x)) -1, we obtain the following boundary condition in the space
~{o; ~}: A(x)
(x-{-i) dHi+--~
(z--i)~,
B(z)
x-f-i
(x--i!~
Hi--~-Gi,
(11)
where
Gt = ( z + i ) - m [( -x - - i )4 B(x) x+ ~
Let Gt • =
(t /2) [•
+
S]GI.
+
A(x-------~)G-/X+] (x + ~)'~
i) ~ H i + - - G l + =
x ~ i ,x,
[
B(x) ( z - - i) B
Hi- -- -----:-. x --
we apply the analog of the generalized IAouville theorem (see [15]). When ~ > 0 we have
434
"
After reducing the boundary condition (117 to the form
A (x)
(z
G+/X-
(z - ~) ~
Gt-
1
A (z) P,,,_~(,) (z + i)~ Ht+ = (z + i)", f- a~+, ~,,_,(z)
B(z) (z -- ~)~ t 1 ~ - =
(x -- i),,, +
(i2)
( x+, ),,, ---'-=x--t
aC,
where Pxl .{x) is a polynomial of degree x I -- 1 with arbitrary coefficients. The problem (11) and the system (12) are equivalent in the sense that at1 solutions of the problem satisfy the system, and conversely all solutions Hi• E La~-{0; co} of the system are solutions of the probtem. Solving the system,we find that, when • > 0, the general solution Hie of the inhomogeneons problem (11) is made up of the general solution of the homogeneous problem: F+--
P gt--i (X)
P cza--i
(x + i) ~'-, ~'~ ~ A~"6('~)+(x-- a.), (13)
F___
P,,,-i(z)
q ~j-~
(~-~)"'-' ~ y,B~,~(')-(:~--b~),
where Aks and BjZ are constants uniquely determined by the conditions A (x)
(z+i)~
F+ ~--_ P,q-i (x)
B (x)
(x + i)~, '
F - -~- 9p ~ ' - i ( x )
(z - i)~
(z-
i)~,
and a particular solution of the inhomogenedus problem, The latter solution is
(z + ~)~'q) (x) -- Q ( z ) ) ( Y+, (1)) ----- Gi +,
A (x)
' (i4)
,,,, z -
~ ,,
B(x)
, '
where QCx) is the Hermite interpolation polynomial (see [16], Sec. 13) of degree a -- 1 for the function (x + i)ar with interpolation points a k of multiplicity r k, and Qi(x) is the corresponding polynomial for (x - i ) $ r The g.f. F s and "/~ are obtained by dividing the right-hand sides of Eq. (12) term by term by (x + i)'ctA(x) and (x - i)" t3B(x), * The explanation of the form of the solution of the homogeneous problem is that, among all g,f. of the space L2• co} the ortly inverse for the function (x - a) k is the g.f. CkS(k-1)i(x - a).T Direct substitution shows that the g.f. (13) and (14) satisfy the boundary condition (11). We show that solutions of the above form exhaust all solutions Hi • E I-e---{0; oo} all solutions of (11). We assume that there exists a g.f. Ha• E/~• co}, Then for any test function {D(z) ~/-~{0; - - c o } we have
A (z) ((z+ 0~/7+' r
=~
of the system (12), and so
satisfying the system (12) and I~s = Has - H2~ ~ 0.
B (z) ((z--,)~ //-, ~(z)) -- O.
We easily see that the g.f. H+ is concentrated at the set of zeros of A(x). Hence Theorem 1 shows that *The operators on the right in (14) are continuous in the space L2{O;*~}, (t ) (--')h-'~_~)(a) " l ' ( 6 c h - ~ ( z - - a ) , ~ ( z ) ) ~ .~[• ~(z) = • 2
435
~h -1
tel+ --- ~ ~ anjS(J)(x -- a~).
From the condition
/Jr ~ L2+{0; co} it follows that akj = 0 (0 ~ j ~ a k -- 1,
h~--~-t j = 0
1 --< k -< p), and so t~+ - 0. We can prove similarly that t~- --- 0. When ~q <-- 0, the solution of (11) exists only when the following
(G,, (x-Fi) "h)
=
0
I ,1 solvability
(k ---- i, 2 . . . .
conditions hold:
(15)
,
If all these conditions are satisfied, the unique solution of the problem has the form (14). We assume that the general solution of (11), and so of (10), has been found. The system (9) with a known right-hand side is now solved in a similar way. From the first equation of (9) it follows that the g.f. F is determined with a precision up to arbitrary g.f. Fa and F c concentrated at the poinl~ of thesets a = {al ..... ap} and c ={c I ..... c/~ with the appropriate orders at these points. The second equation of the system uniquely determines the g.f. Fa by the condition IX 2 + K2(x)JF a = 0. Hence any r
}'1-1
solution of the system (9) is determined with a precision up to a g.f. Fe ~--- ~
ClsS(S)(x-- el)
~
with arbitrary
l~---t s=O
coefficients c/s. * T h e final result for "dual" equations in the space L~{,o; m} can be stated as follows. THEOREM 4. If the functions ~j + Kj(x) (j = 1,2) have common zeros el with multiplicity 7l(1 ~ t -< r), the homogeneous system (7) is solvable in L2{*o; m}; for >q > 0 it has • + 7 linearly independent solutions, while for • - 0 it has
l=t
linearly independent solutions. The inhomogeneous system (7) is solvable with any right-hand side g Q Lz{cx>; ra}, if ~1 -> 0; a necessary and sufficient condition for it to be solvable when n I < 0 is that conditions of the form (15) be satisfied. 4.
Equations
with
Two
Kernels
from
L~{.*;
m}
In spite of the fact that the space L1 {.o; m } has a structure similar to Lz {,o; m}, several of its properties prevent the extension of the reasoning of the previous section to it. Hence in the solution of the Riemann problem in R{0; ,o} we use another approach, suggested by V. S. Rogozhin [17]. We consider an equation with two kernels
t ~kl(t--x)~p(t)dt) --
( /_,
(z) +
t
~k2(t--x)(p(t)dt)=(g,
qp),
where q>(x) ~ L t { - - o o ; - - m ) is an arbitrary test function, kj(x) ~ Lt{0; - - o o } , and ]~ are unknown g.f. from the space LI___{oo, m } .
(16)
the g.f.
g~Lt{oo; ra},
The Fourier transformation of (16) yields the following equivalent Riemann problem in the space R { m ; co}:
[h +
= [x2 +
+
Let the coefficients in the problem satisfy the conditions of the previous section. an important role in the following reasoning, we assume that
(17) Since common zeros do not play
* ([Xzq-K2(z)]Fe, qJ) --~ 0 for any (I) O.L:{--m; --oo}. ~'If the functions Xj + Kj(x) have no common zeros, the solvability of the homogeneous system and the number of its solutions are determined by the index ~i"
436
~, + K,(z) = A (z)R,(z)/(z + 0 5 ~2 + K~(z) = B(z)K,(z) / (z -- ~)~, where all the components of this representation satisfy the former conditions except for the condition a k r bj. We reduce the problem (17) to an equivalent Riemann problem in R{0; ,r
(x 4- i) = F~+ --_ ~ x-4-i ] (x--i)~ where • = • + m, • = Ind [X,//r a, = G/X+~,, ann F,• = F• / (z +_ O~X+-.(x) ~ R~-{0; oo} In the solution of the analogous problem (11) in L2{0; ,o}, we essentially used Theorem I concerning the structure of g.f. concentrated on a set consisting of a finite number of points. We avoid the difficulties inherent tn the proof of a similar theorem in R{0;*r by adopting a different method for the solution of (18). The basic idea of this method is to seek a solution of the auxiliary Riemann problem in the space R{0; --,~} of test functions, such that this solution is continuous in the topology of the space. The required solution in the class of g.f. is constructed by using the boundary condition (18) and the solution of the auxiliary problem. In R{0; --~} we consider the problem
(~ + i)~ - A(x) r
:~ - i )-,,, (z - i) (~--~2 B(x) ~2-(x)-kO(x).
On the right-hand side the function q)(k) ~ R{0; - - o o } . obtained by the method used, for example, in [19].
The solution r
(19)
6 R*-{0; --oo} of (19) can be
We state the following results: a) if x~ < 0, the inhomogeneous problem (19) is solvable for any right-hand side (I)(x) E R {0; - - o o } and has ]~q l linearly independent solutions; the general solution is
,+(x) -- (zA(x) + ~)~ I (P-,,,_~(x) z + i)-,~, ~ r
], (20)
B(x)
(~_,)_,~ * - ( ~ ) = ( x - , ) ~ [. p_.,_~ (x) 4 - ( ~~- T- ,i )~.,o - ( ~ ) ] ; b) if xl - 0, the ~1 necessary and sufficient conditions for its solvability are t*
$ ( x ) (x + i)-h dz = 0
(k ----- t, 2 , . . . , z~).
(21)
and if these conditions are satisfied, the unique solution is given by the formulas
r
A(x)
(z+~)=r
,-(z)--
B(x)
(x-i)~
( x - - i )~'*-(x). z+i
(22)
We now turn to (18). We let x 1 < 0 and try to find the g.f. FI+. Successively using (19), (18), and (20) we find that
(Am,+,. ) = ( (xA(~) + 0" _ (A(x) (z+i)~ -(x.+ i)~ F~+, A (x) --
x+ ~
= (
(x -- ~)~
.-.-. (z + 0-~,/
-7-2-V x--i
*) B(~] --*-(~) ) B(x)
r
).
= (G~+, . ) + y a . (G,, (~ + i)-~).
where the Ck are arbitrary constants.
437
It is plain that the conditions (G,,(x + i) -k) = 0 (k = 1,2 ..... In, l) are necessary and sufficient for the functional on the left-hand side to be continuous in the topology of R{0; --~}. Hence
A(z)
= (Fi§162 = (Fl+, (x +
) =(a,§162
(23)
In the spaceR{0; --*:} the operator A* has a continuous linear inverse, and so the equation A ' r = ~ has a unique solution for arbitrary ~p(x) @ R { 0 ; - - o o } . in fact the required operator is
A._l~; = (x q- i)~p(x)-- q(x), A(~) where Q(x) is the Hermite interpolation polynomial for (x + i ) ~ ( x ) . Hence the unique solution of Eq. (23) is /
(Ft +, r = (Gt+, A ' - l r
=
Gt+,,
(~ + i) ~r (z) - Q (~)
A(x)
)"
The g.f. ( F , ' , ~ ) is constructed similarly. Now let ~1 >- 0. Using the same method but with the relations (22) instead of (20), we find that (AF+,~) = (G,+,~). However the functional (AFI+,r is not defined on the whole space R{0; ---~} but only for functions ff)(x) ~ R { 0 ; - - c o } , satisfying conditions (21). Using a theorem in (17), we construct its extension to the whole space by means of the formula )gl
(AFI+,~) = (Gt+, ~l)(x) )q - ~,.~Ch ~ ~)(t) (t q- i)-kdt h=i
=
a~+ + Y, C~ (z + i)-k, (~ (x)
-r
=
at+ + (z + i) ~,
h=t
where the C k are arbitrary constants and ~'(x) is an arbitrary function of R{0; -"0}. Now
(Fi+,$)=
A-t Gl+q- (xq-i) u' ,ap -~-(Gi+,A'-t~) p r'a-t
+
(~ + i)~,-~ Y~ Y, A~j~(J~-(~ -- ~ ) , /~=i j----o
(there is a similar result for (F,',~)). Thus there are results for the problem (18) similar to those obtained for the problem (11). If the solution F, + of (18) is known, then the solution f~ of (16) can be found by using the inverse Fourier transformation V" 1:
f4- = V ' t [ X •
(x -q- i)mFt+].
(24)
The final result for the equation with two kernels can be stated as follows. THEOREM 5. If • -> 0, Eq. (16) is solvable without conditions and has ~q linear independent solutions; if ~1 < 0, I~11 conditions of the form (21) are necessary and sufficient for its solvability. The general solution can be obtained from (24), where F 1 + gives the genera1 solution of (18~ For classes of solutions in the investigation of the above integral equations w e have chosen the countablynormed spaces Lp{,O; m} (p = 1,2) of the functions increasing at infinity no more rapidly than some polynomial of a certain degree. A peculiarity of such classes is the fact that the special case investigated is found to be normal in the sense that, for a nonnegative index, the integral equation is solvable in these spaces without any further limitations on the free term. 1"It can be proven that A* -i is continuous in R{0; --~}.
438
Even in the spaces Lp{n; m}* the integral operators (1) and (2) are not invertible without extra conditions on the right-hand side. Some sufficient condifionsfor their invertibility for the space L~{n; m} are given in [10J. We note that the corresponding case of integral equations of the convolution type in one subclass of summable functions is investigated in [18]. In conclusion we illustrate the above constructions by two simple examples of solutions of Wieaer--Hopf equations (1):
([+,(p(t)--
a)
{
) ( '-~6(t)+tt]4,~(t),),
t
f eT-'~(x)dx =
vk=
o,
k(t) = --~/~-~et, t < O. ' i g+= ~(t)+It]+~L~+{t;t}cL~+{~;l}, y2~
x --
m---t.
The solution 2c+ is sought in Ll+{*o; I}. The Riemann problem in R{1; ~} is
X x--i
F+
F-~i[~=_~
~z~ 6(i)+(x)1
•
Its unique solution is
F+-=-i [-~z --6+,z, }q-i ~2[6(z)+(x)-- 2,6(i'+(z,]. Hence i /+ = --___I6 (t)
-
-
[i]+]
-
-
+
is the unique solution of the integral equation oO
t
k(t)=
{--]~20e-t,
t >0,~6
,
t < O.) Ll {--oo;O}, K ( x ) =
g+-~-fe2it]+q-~2n6(t-- t ) ~ L ~ + {0; I} c L ~ + {oo; l},
--i
x+i
'
m - - i.
The Riemann problem in R{O; ,0} is
x~F+
xq-i
= F- -4- []/2n"--8+(xq- 2) q- ei=],
•
•
Its general solution, depending on one arbitrary constant, is
F+_~_C[tq_2z6+(x)]..t_
n
(2__i)6+(x+2)q_~_6+(x)
q_ ei=[2n6+(x) q_ i].
The genera1 solution ]+ ~ Li+{O; 1} of the equation is
*If a g.f. f 6 L~{n; m}, in See. 1).
its growth at ,o is bounded by a polynomial of degree n (for an exact definition see above 439
[ (1---~ ' ) eZ"+ h = V-iF+=~2rcC[6(t)+[l]+]+ +~2--~[~(t)+O(t--t)],
+
/9(t_ t)= {t, t>O, 0, t<0,
where C is an arbitrary constant. LITERATURE
1. 2. 3. 4. 5. 6. 7. 8. 9. 1O. 11. 12. 13. 14. 15. 16. 17. 18.
CITED
B. Noble, The Wiener--Hopf Method [Russian translation], IL, Moscow (1962). Yu. I. Chersldi, Problems of mathematical physics reducing to the Riemarm problem, Tr. Tbilissk. matem. in-ta, 28, 209-246 (1962). I.M. Rappoport, A class of singular integral equations, DAN SSSR, 59, 8, 1403-1406 (1948). F.D. Gakhov and Yu. L Cherstdi, Singular integral equations of the convolution type, Izv. Ak. Nauk SSSR, seriya matem., 20....., 1, 33-52 (1956). M.G. Krein, integral equations on a half-line with kernels depending on the difference between the argumenu, Uspekhi matem, nauk, 13, 5, 3-120 (1988). G.A. Dzhanashiya, Convolution type equation for a semiaxis with bounded right-hand sides, Soobshch. Ak. Nauk GruzSSR, 36, 1, 11-18 (1964). V.V. Ivanov, The Wiener-Hopf equation of the first kind, DAN SSSR, 151, 3, 489-492 (1963). O.S. Parasyuk, Dual integral equations in the class of generalized functions, DAN SSSR, 110, 6, 957-958
(1956). Yu. I. Cherskii, Integral Equations of the Convolution Type with Applications [in Russian], Doctoral Dissertation, Tbilisi (1964). V.B. Dybin, An exceptional case of integral equations of convolution type in a space of generalized functions, DAN SSSR, 161, 4, 753-756 (1965). F.D. Berkovich, An integral equation on a semiaxis in a cIass of functions increasing toward infinity, DAN SSSR, 160, 2, 255-258 (1965). I . M . Gel'land and G. E. Shflov, Generalized Functions[in Russian], Fizmatgiz, Moscow (1956), Vol. 2. R.D. Banuuri and G. A. Dzhanashiya, Convolution type equations on a semiaxi~, I)AN SSSR, 155, 2, 251-253 (1964). F . D . Gakhov, Boundary-Value Problems [in Russian], Fizmatgiz, Moscow (1963). Yu. I. Cherstdi, A contribution to the solution of the ~ e m a n n boundary-value problem in classes of generalized functions, DAN SSSR, 125, 3, 500-503 (1959). V.L. Goncharov, Interpolation Theory and the Approximation of Functions [in Russian], Fizmatgiz, Moscow (1954). V.S. Rogozhin, A general scheme for the solution of boundary-value problems in a space of generalized functions, DAN SSSR, 164, 2, 277-280 (1965). F.D. Gakhov and V. L Smagina, A special case of integral equations of the convolution type and equations of the first kind, Izv. Ak NaukSSSR, setiya matem., 26, 3, 361-390 (1962).
All abbreviations of periodicals it, the above bibliography are letter-by-letter transliterations of the abbreviations as given in the original Russian journal S o m e or all of this peri. o d i c a l l i t e r a t u r e m a y w e l l be a v a i l a b l e in E n g l i s h t r a n s l a t i o n . A complete l i s t of the cover-tocover English translations appears at the back of the first issue of this year.
440
